Generalized Rich-Club Ordering in Networks

Rich-club ordering refers to tendency of nodes with a high degree to be more interconnected than expected. Such kind of ordering can be quantitatively recognized via the coefficient φ(k)=(2E_(>k))/(N_(>k) (N_(>k)-1)) where E_(>k) is the number of links among the N_(>k) nodes having degree higher than a given value k and (N_(>k) (N_(>k)-1))/2 is the maximum possible number of links among the N_(>k) nodes [1]. The rich-club coefficient, when compared with its expectation, 〖φ(k)〗_rand, over a set of rewired networks with the same degree sequence of the original one, is called 〖φ(k)〗_norm and a network is said to display rich-club ordering when 〖φ(k)〗_norm>1 [2].
Networks are, in general, characterized by both structural and non-structural patterns thus, the concept of rich-club ordering can be easily generalized to structural measures different from the node degree as well as to non-structural measures (i.e. to node metadata). The differences in considering rich-club ordering (RCO) with respect to both structural and non-structural measures depends on the employed coefficients and on the appropriate null models (link rewiring vs metadata reshuffling) that we choose to adopt. In the case of structural measures different from the node degree rich-club ordering can be evaluated by creating a ranking of nodes or, in other words, by considering the rich-club coefficient as a measure of position. This is because, in such a case, by using the rewiring procedure the structural measure of node i and the number of node retaining such a measure may change. If we assign each node to a position p, then rich-club ordering in the case of structural measures different from the node degree can be evaluated with the coefficient φ(p) and then normalized over the ensemble of rewired networks.
In the case of non-structural attributes (i.e. node metadata) the link rewiring doesn't seem to be the unique option for generating a null distribution. Indeed, we may be interested in knowing if different arrangements of the node metadata over the same network structure are able to unveil rich-club ordering as well. In other words, we may also be interested in using a null model that keeps the original network structure while reshuffling the node metadata. More intuitively, when we evaluate rich-club ordering with the links rewiring we are basically asking the question: does the considered network possess a topology so unusual that it allows room for rich-club ordering? While if we evaluate rich-club ordering with metadata reshuffling we are basically asking the question: does the considered network possess an arrangement of node metadata so unusual that it allows room for rich-club ordering?
In the case of node metadata the rich-club coefficient can be easily derived from the case of node degree just considering, instead of the degree k, a certain value m corresponding to the value of the node metadata. The normalized rich-club coefficient, 〖φ(m)〗_norm, can be derived considering 〖φ(m)〗_rand from two different perspectives that depend on the null model that we use. In the case of link rewiring, we use the coefficient 〖φ(m)〗_norm^rew= (φ(m))/(〖φ(m)〗_rand^rew ) while in the case of metadata reshuffling we use the coefficient 〖φ(m)〗_norm^resh= (φ(m))/(〖φ(m)〗_rand^resh ). In summary, by considering different notions of node richness we can quantify structural and non-structural rich-club ordering, observing how external information about the network nodes could be able to validate the presence of élites in networked systems.